# Analyis I: Convergence, Elementary Functions

###### Roger Godement
Publisher:
Springer Verlag
Publication Date:
2004
Number of Pages:
430
Format:
Paperback
Series:
Universitext
Price:
59.95
ISBN:
3-540-05923-7
Category:
Textbook
[Reviewed by
Michael Berg
, on
06/13/2006
]

To say Analysis I and Analysis II  are idiosyncratic is not to do them justice. Roger Godement has given us a unique analysis text, bursting at the seams with beautiful and serious mathematics presented in a very unusual way and serving at the same time as a venue for historical commentary and extensive criticism of e.g. the American military-industrial complex.

Perhaps it is correct to say that Godement’s subtext, or, rather, psychological motivation for this work was to indulge a pair of desires: to address the minutiae of analysis at the undergraduate level (covering just about everything short of Lebesgue theory, except for a short introduction in the Appendix to chapter V) and to open the flood gates vis à vis the author’s strong feelings about a variety of themes cherished by the French political left. So, predictably, Godement trains his cross hairs on such matters as the birth and evolution of American weapons research from the Manhattan Project to Star Wars, all abetted by extensive lists of figures and a plethora of references to recent works by historians and sociologists, mainly French and American (but not exclusively so). He also discusses, again at considerable length, the history of Soviet secret research on germ and biological weapons, capped off by the flight to freedom (i.e. the United States ) of some of its major figures and the dispersion of many of the existing strains of deadly viruses to rogue states and terrorists.

Godement’s objective in presenting the reader with this material, most of which is found in the long Postface to Analysis II, is to warn the reader, i.e. the young mathematics student at the start of his professional path, against the seductive temptations of certain venues of applied research, or even basic research prone to military application; in this connection Godement’s Postface (already foreshadowed in his lengthy Preface to Part I) starts with a section titled, “How to fool young innocents,” manifestly referring to the intended audience itself.

Godement’s historical asides, of varying length and frequency (Parts I and II are rather different: Part II is, modulo its Postface, less politically polemical and therefore more mathematically orthodox, so to speak), are similarly well-researched and colored by the author’s personal views; of course, when it comes to marginalia to what is fundamentally a very serious mathematics book (rather than a historical essay) it cannot be otherwise. The reader will in any case find a lot of marvelous material about the earlier analysts, and Euler is particularly well-represented.

In this connection I should like to draw attention to two particulars: on p. 94 of Part I Godement, apparently a true son of the French Enlightenment, indicts Frederick II of Prussia for choosing Euler to tutor his niece, instead of Diderot, d’Alembert, or Voltaire; he goes on to suggest that Euler, as a devout Calvinist, should be regarded as anomalous: “…his father was a pastor… and it was mathematics which without changing his ideas eventually dissuaded the young Euler from following him.” Unfortunately the timbre of this remark is not atypical. On the other hand, it is of course impossible to doubt Euler’s titanic gifts (and Godement properly refers to him as “the Bach of mathematics”) and we find, on p. 413, the following passage: “…since everything was possible to people like Euler it is possible that he would have delivered himself to [the exercise of computing the coefficients of the generating function for the partition function] since electronics and computer science were not yet, in his age, the two breasts at which advanced Humanity nourished itself, nor the milliard of breasts at which these two industries nourish themselves…” [sic]. (À propos, harmless grammatical errors like those in this passage occur from time to time, but they are almost of infinitesimal significance.)

On to the mathematics, then, and in a word, it’s superb. Analysis I and Analysis II, actually the first of four books, are encyclopedic in scope and are filled with marvelous and expansive expositions. The subtitle of Analysis I, running to over 400 pages, is “Convergence, Elementary Functions,” and that of Analysis II, at about the same number of pages, is “Differential and Integral Calculus, Fourier Series, Holomorphic Functions.” And consider, e.g., the list of topics in the chapter on differential and integral calculus: the Riemann integral, integrability conditions, the Fundamental Theorem, integration by parts, Taylor ’s formula, the change of variable formula, generalized Riemann integrals, approximation theorems, Radon measures on R or C, and Schwartz distributions. This arrangement illustrates the fact that the goal of this series of books is to present introductory analysis in its natural, organic sequence of topics, with each given far more air-play than is customary in ordinary undergraduate programs.

It is not Godement’s goal to develop a prerequisite topic only to the point necessary for what might come next, all in obedience to a pre-ordained syllabus. No, it is Godement’s goal to present a lot of wonderful mathematics — in context — for its own sake, and because young (or even not so young) mathematicians should know it. Accordingly the long books read like a conversation with an expert who not only loves his subject but talks about it brilliantly.

While these books should probably be kept far away from a randomly chosen beginning upper division student, lest he fail to stay “on sequence,” they are truly magnificent sources for a deeper study of analysis. In another order of Providence, our randomly chosen student would be exposed to a less utilitarian curriculum and would learn analysis in the order and manner presented by Godement. Indeed, it is possible to draw a close comparison between the books under review and the classic, A Course of Pure Mathematics, by G. H. Hardy, which also presupposes considerable mathematical maturity on the part of the reader. By contrast, however, Hardy’s style could only be characterized as concise.

I first came across Roger Godement’s work in Topologie Algébrique et Theorie des Faisceaux, which, while dated, is still a fantastic introduction to the theory of sheaves, very well-written, very complete, and very rigorous. The same is true about the mathematics in Analysis I and II and is a pure pleasure to read and study. The reader, whoever he might be, whether an advanced undergraduate or a veteran, will finish these pages with a deeper understanding of the subject and an increased sense of mathematical culture.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

Preface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : V

I – Sets and Functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1

x1. Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1 – Membership, equality, empty set . . . . . . . . . . . . . . . . . . . . . . 7

2 – The set defined by a relation. Intersections and unions . . . 10

3 – Whole numbers. Infinite sets . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 – Ordered pairs, Cartesian products, sets of subsets . . . . . . . 17

5 – Functions, maps, correspondences . . . . . . . . . . . . . . . . . . . . . 19

6 – Injections, surjections, bijections . . . . . . . . . . . . . . . . . . . . . . 23

7 – Equipotent sets. Countable sets . . . . . . . . . . . . . . . . . . . . . . . 25

8 – The different types of infinity . . . . . . . . . . . . . . . . . . . . . . . . . 28

9 – Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

x2. The logic of logicians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

II – Convergence: Discrete variables : : : : : : : : : : : : : : : : : : : : : : : : : : 45

x1. Convergent sequences and series . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

0 – Introduction: what is a real number? . . . . . . . . . . . . . . . . . . 45

1 – Algebraic operations and the order relation: axioms of R . 53

2 – Inequalities and intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 – Local or asymptotic properties . . . . . . . . . . . . . . . . . . . . . . . . 59

4 – The concept of limit. Continuity and differentiability . . . . 63

5 – Convergent sequences: definition and examples . . . . . . . . . . 67

6 – The language of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7 – The marvels of the harmonic series . . . . . . . . . . . . . . . . . . . . 81

8 – Algebraic operations on limits . . . . . . . . . . . . . . . . . . . . . . . . 95

x2. Absolutely convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9 – Increasing sequences. Upper bound of a set of real numbers 98

10 – The function log x. Roots of a positive number . . . . . . . . . 103

11 – What is an integral? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

12 – Series with positive terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

13 – Alternating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

14 – Classical absolutely convergent series . . . . . . . . . . . . . . . . . 123

15 – Unconditional convergence: general case . . . . . . . . . . . . . . . 127

XX Contents

16 – Comparison relations. Criteria of Cauchy and d’Alembert 132

17 – Infinite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

18 – Unconditional convergence: associativity . . . . . . . . . . . . . . 139

x3. First concepts of analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . 148

19 – The Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

20 – The principle of analytic continuation . . . . . . . . . . . . . . . . . 158

21 – The function cot x and the series P1=n2k . . . . . . . . . . . . . 162

22 – Multiplication of series. Composition of analytic functions.

Formal series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

23 – The elliptic functions of Weierstrass . . . . . . . . . . . . . . . . . . 178

III – Convergence: Continuous variables : : : : : : : : : : : : : : : : : : : : : : 187

x1. The intermediate value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

1 – Limit values of a function. Open and closed sets . . . . . . . . 187

2 – Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3 – Right and left limits of a monotone function . . . . . . . . . . . . 197

4 – The intermediate value theorem . . . . . . . . . . . . . . . . . . . . . . . 200

x2. Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5 – Limits of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . 205

6 – A slip up of Cauchy’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7 – The uniform metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

8 – Series of continuous functions. Normal convergence . . . . . . 220

x3. Bolzano-Weierstrass and Cauchy’s criterion . . . . . . . . . . . . . . . . . 225

9 – Nested intervals, Bolzano-Weierstrass, compact sets . . . . . 225

10 – Cauchy’s general convergence criterion . . . . . . . . . . . . . . . . 228

11 – Cauchy’s criterion for series: examples . . . . . . . . . . . . . . . . 234

12 – Limits of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

13 – Passing to the limit in a series of functions . . . . . . . . . . . . 241

x4. Differentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

14 – Derivatives of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

15 – Rules for calculating derivatives . . . . . . . . . . . . . . . . . . . . . . 252

16 – The mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

17 – Sequences and series of differentiable functions . . . . . . . . . 265

18 – Extensions to unconditional convergence . . . . . . . . . . . . . . 270

x5. Differentiable functions of several variables . . . . . . . . . . . . . . . . . . 273

19 – Partial derivatives and differentials . . . . . . . . . . . . . . . . . . . 273

20 – Differentiability of functions of class C1 . . . . . . . . . . . . . . . 276

21 – Differentiation of composite functions . . . . . . . . . . . . . . . . . 279

22 - Limits of differentiable functions . . . . . . . . . . . . . . . . . . . . . 284

23 – Interchanging the order of differentiation . . . . . . . . . . . . . . 287

24 – Implicit functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

0 Contents

Appendix to Chapter III : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 303

1 – Cartesian spaces and general metric spaces . . . . . . . . . . . . . 303

2 – Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

3 – Limits and Cauchy’s criterion in a metric space; complete

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

4 – Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

5 – Absolutely convergent series in a Banach space . . . . . . . . . 313

6 – Continuous linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

7 – Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

8 – Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

IV – Powers, Exponentials, Logarithms, Trigonometric Functions

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 325

x1. Direct construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

1 – Rational exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

2 – Definition of real powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

3 – The calculus of real exponents . . . . . . . . . . . . . . . . . . . . . . . . 330

4 – Logarithms to base a. Power functions . . . . . . . . . . . . . . . . . 332

5 – Asymptotic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

6 – Characterisations of the exponential, power and logarithmic

functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

7 – Derivatives of the exponential functions: direct method . . 339

8 – Derivatives of exponential functions, powers and logarithms342

x2. Series expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

9 – The number e. Napierian logarithms . . . . . . . . . . . . . . . . . . . 345

10 – Exponential and logarithmic series: direct method . . . . . . 346

11 – Newton’s binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

12 – The power series for the logarithm . . . . . . . . . . . . . . . . . . . 359

13 – The exponential function as a limit . . . . . . . . . . . . . . . . . . . 368

14 – Imaginary exponentials and trigonometric functions . . . . 372

15 – Euler’s relation chez Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

16 – Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

x3. Infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

17 – Absolutely convergent infinite products . . . . . . . . . . . . . . . 394

18 – The infinite product for the sine function . . . . . . . . . . . . . . 397

19 – Expansion of an infinite product in series . . . . . . . . . . . . . . 403

20 – Strange identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

x4. The topology of the functions Arg(z) and Log z . . . . . . . . . . . . . 414

Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 425